In the sense that I made it through the eight videos I was hoping to get through this week, I achieved my goal, however I didn’t completely understand the last vid or make notes on it so it didn’t really count… Nonetheless, it was still a fairly productive week! Compared to the past few weeks, I had a decent understanding of most of what was being talked about. I got a better idea of how rotation matrices work and the notation used in linear algebra in general. As often is the case, I wish I would have gotten more done this week but the notes I took were pretty legit, so I’ve at least got that going for me which is nice. So, ya. Not a great week, but it could have been worse. 😬
(I’ve got a million things happening right now, so I’m not going to write much for any of the videos. 🙃)
Video 1 – Sums and Scaler Multiples of Linear Transformations
In this video, Sal explained that the same rules that apply to linear transformations on vectors also apply to the transformations themselves. (I.e. you can add matrix transformations together and scale them in the same way you can add and scale vectors together before and after transformations.) I should mention that the notes I wrote out above don’t show the proof for scaling transformations which was covered at the end of this video.
Video 2 – More on Matrix Addition and Scaler Multiplication
This video worked through an example of what Sal was talking about in the previous one.
Video 3 – Linear Transformation Examples: Scaling and Reflections
I think my notes do a decent job of explaining what this video was talking about so I won’t elaborate, BUT it was cool to see how you can rotate shapes using linear algebra which Sal said is the essence of how computer graphics work (although I’m sure this is a rudimentary version of it). I will also note that this video helped my understanding of the notation and also with doing matrix multiplication.
Video 4 – Linear Transformation Examples: Rotations in R2
I don’t have time to fully explain how it works, but here Sal shows how to use trig to rotate vectors in 2D. In essence, you take the identity matrix of R2 and replace the columns with the trig matrix shown in the bottom left corner of the second screenshot above. The reason why this works has to do with the unit circle and taking the vector that was laying on [1, 0] (which you should think of as the hypotenuse of a right triangle) and rotating it around to a given angle which would then set the vector’s new coordinates to be [cos(θ), sin(θ)]. You do the same thing for the y-basis vector, [0, 1]. You multiply whatever vector you’re looking at by the trig matrix and, boom, you’ve rotated that vector.
Video 5 – Rotation in R3 around the x-axis
In this vid Sal shows how rotation in 3D around the x-axis turns out to be essentially the same type of rotation matrix as it is in 2D. I’m not 100% sure, but I’m guessing the matrices for rotations around each axis would be the following:
Video 6 – Unit Vectors
I don’t think the concept Sal explained in this video, calculating a unit vector, is too difficult to understand, but the formula hasn’t sunk in for me completely. The steps it takes to derive the formula seems pretty straightforward, but I can’t easily visualize them all coming together unanimously.
Video 7 – Introduction to Projections
Like the previous video, I have a hard time grasping all of what Sal was talking about in this video in my head at the same time. I can follow along with what’s going on and can understand each step relatively easily, but I find it hard to visualize what’s happening in its entirety at the same time. Nonetheless, I know that this video is explaining how to find ‘c’, the scaler used to scale a vector along a line to a perpendicular drawn on that line leading to another vector, a.k.a. the point at which another vector is being projected onto the line. It’s confusing but at least I understand what Sal is talking about even if I don’t clearly and intuitively grasp how and why all the math comes together.
Like I said at the start, I watched the final video in this lesson section but will need to rewatch it again at the start of this coming week. So get excited for that. 🥳
I’m unfortunately not really that close to finishing off this unit. I also have a lot of things on the go these days – one of which I’ve referred to a handful of times over the past few weeks which I’m REALLY excited about, but which is taking up a ton of my time – so it may take me awhile before I manage to get to the next unit. If I could get to it by the start of the New Year that would be good. If that’s the case, I’ll be disappointed thatI didn’t get further, but having that as a goal is probably what’s most realistic. For the thousandth, billionth time, I KNOW I’m going to get through this all, but it’s really starting to feel like it will never end… 😮💨